Essex Dogs by Dan Jones describes in passing English or Welsh archers 1 using thumb rings during the hundred year's war.
I can find (unreliable) references to Roman thumb rings, and certainly the Romans encountered Asiatic archers, so it is not implausible. And Western European archers would have encountered Huns (plus the crusades). It is quite plausible.
But I've found no record of Asiatic/thumb draw in any of my sources on English Archery. I'm reluctant to conclude that absence of evidence is evidence of absence.
Is there a fundamental reason why Western/English archers preferred the Mediterranean draw?
1 I listened to it in an audio book which makes it very difficult to go back and check. I think the reference is to the English archer, but even the Welsh archers are fighting (most of the time) in support of the English army. It is also possible that I misunderstood a reference to an ornamental ring as a reference to an archer's thumb ring.